Trigonometry for Year 9 — sine, cosine and tangent

Trigonometry is the branch of maths that deals with the relationships between the sides and angles of a triangle. In Year 9 you meet it for the first time through the right-angled triangle, and learn three basic functions: sine, cosine and tangent. This guide explains from the ground up what each of them means, how to calculate them, how to remember the formulae with SOH-CAH-TOA, and presents three full examples including finding the height of a tree, finding an angle from two sides, and a ladder leaning against a wall.

Trigonometry is the branch of maths that studies the relationships between the sides and angles of a triangle. In Year 9 you learn the foundation of the subject through the right-angled triangle, and meet three central functions: sine (sin), cosine (cos) and tangent (tan). These three functions are really just simple ratios between two sides, and with them you can find missing sides, calculate unknown angles, and solve geometry problems that cannot be solved with Pythagoras' theorem alone.

The guide explains every idea from the beginning, presents full numerical examples, and suits anyone meeting the topic for the first time.

What is trigonometry?

The word 'trigonometry' comes from Greek: trigonon (triangle) and metron (measure). Literally: 'measuring triangles'. That is exactly what it does — it links the angles of a triangle to the lengths of its sides.

The basic idea is this: if you look at a right-angled triangle with a particular angle α (alpha — a Greek letter used for an angle), the ratio between any two sides stays the same — regardless of the size of the triangle. A right-angled triangle with a 30° angle will always give the same ratios between its sides, whether it is as small as a matchstick or the size of a building.

These fixed ratios are called trigonometric functions, and in Year 9 you focus on three: sine, cosine and tangent.

The right-angled triangle and its sides

Before defining the three functions, you need to know what the sides of a right-angled triangle are called. A right-angled triangle is a triangle in which one of the angles equals 90° (a right angle).

The three sides are named like this:

• **Hypotenuse** — the longest side, the one opposite the right angle (90°). The hypotenuse is always fixed and does not depend on which angle you choose.
• **Opposite** — the side that is opposite the angle α you have chosen to look at.
• **Adjacent** — the side that touches the angle α (but is not the hypotenuse).

Note: 'opposite' and 'adjacent' swap over depending on which angle you choose. If you look at the other (non-right) angle, the opposite becomes the adjacent and vice versa. The hypotenuse is always the hypotenuse.

The three ratios: sine, cosine and tangent

For every angle α in a right-angled triangle we define three ratios:

• **Sine (sin)**: the ratio of the opposite side to the hypotenuse. sin(α) = opposite / hypotenuse
• **Cosine (cos)**: the ratio of the adjacent side to the hypotenuse. cos(α) = adjacent / hypotenuse
• **Tangent (tan)**: the ratio of the opposite side to the adjacent side. tan(α) = opposite / adjacent

A numerical example: in a right-angled triangle where the angle α equals 30°, the hypotenuse is 10 cm long, the opposite side is 5 cm, and the adjacent side is about 8.66 cm. We get:

• sin(30°) = 5 / 10 = **0.5**
• cos(30°) = 8.66 / 10 = **0.866**
• tan(30°) = 5 / 8.66 = **0.577**

These are the fixed values of the functions for an angle of 30°, and they come up in every right-angled triangle with that angle.

How to remember them — the SOH-CAH-TOA mnemonic

The standard way, used in schools all over the world, to remember the three definitions is the acronym:

• **SOH** — Sin = Opposite / Hypotenuse
• **CAH** — Cos = Adjacent / Hypotenuse
• **TOA** — Tan = Opposite / Adjacent

Some teachers use a formula triangle for each ratio — cover the quantity you want and the triangle shows you whether to multiply or divide. Most pupils say SOH-CAH-TOA out loud two or three times before a test, and it goes into the head for good.

Three step-by-step examples

Example 1: calculating the height of a tree

You stand 20 metres from the base of a tree. The angle from the ground up to the top of the tree is 40°. How tall is the tree?

What is known and what are we looking for: the side adjacent to the angle (the distance along the ground) = 20 metres. The opposite side (the height) = ?

Which ratio fits? We have the adjacent, we want the opposite → use tangent (TOA).

tan(40°) = height / 20. From the calculator: tan(40°) ≈ 0.839. So: 0.839 = height / 20. Multiply both sides by 20: height = 0.839 × 20 = 16.78 metres.

Example 2: finding an angle from two sides

In a right-angled triangle the side opposite the angle α is 7 cm, and the hypotenuse is 14 cm. What is the size of the angle α?

What is known: opposite = 7, hypotenuse = 14 → use sine (SOH).

sin(α) = 7/14 = 0.5

Now we need to 'undo' the sine — to ask: which angle gives a sine of 0.5? This operation is called the inverse sine (arcsin), and on the calculator the button is sin⁻¹.

α = sin⁻¹(0.5) = 30°

The same principle works with cos⁻¹ (when you know the adjacent and the hypotenuse) and tan⁻¹ (when you know the opposite and the adjacent).

Example 3: a ladder leaning against a wall

A 5-metre ladder leans against a wall. The foot of the ladder is 1.5 metres from the wall. At what angle does the ladder rest on the floor?

Analysis: the ladder itself is the hypotenuse (5 metres). The distance from the wall is the side adjacent to the angle with the floor (1.5 metres). We have the adjacent and the hypotenuse → cosine (CAH).

cos(α) = 1.5/5 = 0.3. α = cos⁻¹(0.3) ≈ 72.5°. The ladder rests at an angle of about 72.5° to the floor (a safe angle for a ladder is considered to be between 70° and 75°).

For more calculations you can use the interactive trigonometry calculator, which carries out these calculations automatically and also draws a diagram of the triangle.

The values of the important angles (table)

There are five angles worth knowing by heart — they come up again and again in exercises and tests, and on the GCSE non-calculator paper you are expected to know their exact values:

A few things worth noticing:

• **Symmetry between sin and cos**: sin(30°) = cos(60°), and also sin(60°) = cos(30°). That is no accident — they are complementary angles (they add up to 90°).
• **At 45° sin equals cos** — because the triangle is isosceles and right-angled, and the two shorter sides are equal.
• **tan(90°) is undefined** — because by definition you have to divide by the adjacent side, and when the angle is 90° the adjacent side is 0, and dividing by zero is not allowed.

The link to Pythagoras

There is a central identity that links the sine and cosine of the same angle: sin²(α) + cos²(α) = 1

The explanation is simple: by Pythagoras' theorem, (opposite)² + (adjacent)² = (hypotenuse)². Divide both sides by (hypotenuse)², and you get the identity. Check with 30°: (0.5)² + (0.866)² = 0.25 + 0.75 = 1. ✓

To refresh Pythagoras' theorem itself, you can use the Pythagoras calculator.

Trigonometry in real life

Trigonometry is not just material for a test — it is at the heart of a great many professional fields:

• **Navigation and flying** — pilots calculate flight angles, latitude and headings using trigonometry.
• **Architecture and civil engineering** — calculating roof pitches, the length of beams, the stability of structures.
• **Radar and sonar** — calculating the distance and direction of objects from the angle and the time it takes a signal to return.
• **Computer graphics and video games** — rotating characters, calculating shadows, 3D cameras — all sin and cos functions.
• **Astronomy** — measuring distances between stars, calculating satellite orbits.
• **Medicine** — imaging scans such as CT and MRI use trigonometry to build a 3D picture from slices.

So the question 'when will I ever need this in life?' has a clear answer: even if you are not going to be a pilot, the map app on your phone, the film at the cinema, and even physiotherapy — all run on trigonometry.

Common mistakes

The five mistakes that come up most in Year 9 tests:

1. **Mixing up the sides** — pupils forget which side is 'opposite' and which 'adjacent'. The tip: circle the angle α in the triangle, then 'opposite' is the side that does not touch the circle, and 'adjacent' is the side that touches it but is not the hypotenuse.

1. **Using the wrong function** — before you start calculating, ask yourself: 'Which two sides do I have? Which one am I looking for?' and choose sin / cos / tan accordingly.

1. **Forgetting to use the inverse function** — when you are looking for an angle (not a side), you must apply sin⁻¹ / cos⁻¹ / tan⁻¹ and not sin / cos / tan. If a pupil gets an angle of 0.5° instead of 30° — that is a sign they forgot to press sin⁻¹ on the calculator.

1. **Degrees mode (DEG) versus radians (RAD) on the calculator** — in Year 9 you always work in degrees. If you get strange results, check that the calculator is in **DEG** mode.

1. **Rounding too early** — if you round sin(40°) to 0.8 instead of 0.839, the final result can be out by whole metres. Round only at the end.

← Back to all articles